Black hole evaporation
30Es3
Essays
A brief analysis of the mathematical results of Hawking radiation.

Black hole evaporation.
The Hawking temperature T of a Schwarzschild (nonrotating, uncharged) black hole with mass m is given by the equation (in geometrized units) [reference 1]
T = hbar/(8 pi k m). equation 1
In conventional units (which we use here), this would be written
T = (hbar c3)/(8 pi G k m). equation 2
The emission of this energy results in an energy decrease of the black hole, and thus a loss in its mass. What period of time tau will it take for a black hole of mass mu to evaporate completely?

A black hole with mass m has a Schwarzschild radius

r = 2 G m/c2 equation 3
and thus an area of
A = 4 pi r2 equation 4
A = 16 pi G2 m2/c4. equation 5
Hawking radiation would have a power P related to the hole's area A and its temperature T by the blackbody power law (with e = 1),
P = sigma A T4 equation 6
P = (sigma hbar4 c8)/(256 pi3 G2 k4 m2) equation 7
or more conveniently,
P = K/m2 equation 8
where K == (sigma hbar4 c8)/(256 pi3 G2 k4) = 3.563 x 1032 W kg2. Given that the power of the Hawking radiation is the rate of energy loss of the hole, we can write
P = -dE/dt. equation 9
Since the total energy E of the hole is related to its mass m by Einstein's mass-energy formula,
E = m c2 equation 10
we can then rewrite P = -dE/dt as
P = -(d/dt) (m c2) equation 11
P = -c2 dm/dt. equation 12
We can then equate this to our above expression for the power, P = K/m2, and find
-c2 dm/dt = K/m2. equation 13
This differential equation is separable, and we can write
m2 dm = -K/c2 dt. equation 14
Integrating over m from mu (the initial mass of the hole) to zero (complete evaporation), and over t from zero to tau, we find that
tau = c2/(3 K) mu3. equation 15
That is, the evaporation time of the hole is proportional to the cube of its mass.

References.
1.
Black holes, white dwarfs, and neutron stars: The physics of compact objects
Stuart L. Shapiro, Saul A. Teukolsky
p. 366
Wiley-Interscience; 1983
reference 1
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